Sublinear Time Algorithms

We have long considered showing the existence of a linear
time algorithm for a problem to be the gold standard
of achievement. Indeed, it is hard to imagine doing much
better than that, since for any nontrivial problem, it would seem
that an algorithm must consider all of the input in order to make
a decision. However, as extremely large data sets grow
more prevalent in a wide variety of settings, it is natural
to wonder what one can do in sublinear time. In fact,
there has been a lot of recent interest in this direction.

Sublinear time is a daunting goal since it allows one to read only
a miniscule fraction of the input. There are problems for
which deterministic exact sublinear time
algorithms are known. However, for most natural problems the
algorithm must use randomization and must give an answer which is in
some sense approximate.
There is a growing body of work aimed at finding sublinear
time algorithms for various problems.
Recent results have shown that there are classical
optimization problems whose values can be approximated in
sublinear time. In addition, property testing , an alternative
notion of approximation for decision problems, has been
applied to give sublinear algorithms for a wide variety of
problems.
One can also
test various properties of distributions, where access to the distribution
is given through samples generated according to the distribution, in
time sublinear in the size of the support of the distribution.

Many examples of problems that can be solved in sublinear time
have been found. Several useful
techniques, including the use of the Szemeredi Regularity lemma
and low rank approximations of matrices, have emerged for
designing sublinear algorithms.
Still, the study of sublinear time algorithms is
very new, and much remains to be understood about their scope,
both in finding sublinear time algorithms for new problems
and in finding more efficient algorithms for problems that
already have sublinear time algorithms.

The following contains some pointers to a few surveys on sublinear algorithms
and property testing, as well as slides from a short introductory talk.
These pointers are a bit older.